bug333 (missed optimization)

Average Error: 58.5 → 0.2

Time: 19.9s

Precision: binary64

Cost: 14016

\[-1 \leq x \land x \leq 1\]

Math

\[\sqrt{1 + x} - \sqrt{1 - x} \]

↓

\[0.0546875 \cdot {x}^{5} + \left(0.0322265625 \cdot {x}^{7} + \left(0.125 \cdot \left(\left(x \cdot x\right) \cdot x\right) + x\right)\right) \]

FPCore

(FPCore (x) :precision binary64 (- (sqrt (+ 1.0 x)) (sqrt (- 1.0 x))))

↓

(FPCore (x)
 :precision binary64
 (+
  (* 0.0546875 (pow x 5.0))
  (+ (* 0.0322265625 (pow x 7.0)) (+ (* 0.125 (* (* x x) x)) x))))

C

double code(double x) {
	return sqrt((1.0 + x)) - sqrt((1.0 - x));
}

↓

double code(double x) {
	return (0.0546875 * pow(x, 5.0)) + ((0.0322265625 * pow(x, 7.0)) + ((0.125 * ((x * x) * x)) + x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((1.0d0 + x)) - sqrt((1.0d0 - x))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.0546875d0 * (x ** 5.0d0)) + ((0.0322265625d0 * (x ** 7.0d0)) + ((0.125d0 * ((x * x) * x)) + x))
end function

Java

public static double code(double x) {
	return Math.sqrt((1.0 + x)) - Math.sqrt((1.0 - x));
}

↓

public static double code(double x) {
	return (0.0546875 * Math.pow(x, 5.0)) + ((0.0322265625 * Math.pow(x, 7.0)) + ((0.125 * ((x * x) * x)) + x));
}

Python

def code(x):
	return math.sqrt((1.0 + x)) - math.sqrt((1.0 - x))

↓

def code(x):
	return (0.0546875 * math.pow(x, 5.0)) + ((0.0322265625 * math.pow(x, 7.0)) + ((0.125 * ((x * x) * x)) + x))

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 + x)) - sqrt(Float64(1.0 - x)))
end

↓

function code(x)
	return Float64(Float64(0.0546875 * (x ^ 5.0)) + Float64(Float64(0.0322265625 * (x ^ 7.0)) + Float64(Float64(0.125 * Float64(Float64(x * x) * x)) + x)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 + x)) - sqrt((1.0 - x));
end

↓

function tmp = code(x)
	tmp = (0.0546875 * (x ^ 5.0)) + ((0.0322265625 * (x ^ 7.0)) + ((0.125 * ((x * x) * x)) + x));
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(N[(0.0546875 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0322265625 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.125 * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	58.5
Target	0.0
Herbie	0.2

\[\frac{2 \cdot x}{\sqrt{1 + x} + \sqrt{1 - x}} \]

Derivation

1. Initial program 58.5

   \[\sqrt{1 + x} - \sqrt{1 - x} \]

2. Taylor expanded in x around 0 0.2

   \[\leadsto \color{blue}{0.0546875 \cdot {x}^{5} + \left(0.0322265625 \cdot {x}^{7} + \left(0.125 \cdot {x}^{3} + x\right)\right)} \]

3. Applied egg-rr 0.2

   \[\leadsto 0.0546875 \cdot {x}^{5} + \left(0.0322265625 \cdot {x}^{7} + \left(0.125 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} + x\right)\right) \]

Alternatives

Alternative 1
Error	0.2
Cost	13504

\[\left({\left(0.5 \cdot x\right)}^{3} + {x}^{5} \cdot 0.0546875\right) + x \]

Alternative 2
Error	0.2
Cost	7296

\[0.0546875 \cdot {x}^{5} + \left(0.125 \cdot \left(\left(x \cdot x\right) \cdot x\right) + x\right) \]

Alternative 3
Error	0.3
Cost	576

\[0.125 \cdot \left(\left(x \cdot x\right) \cdot x\right) + x \]

Alternative 4
Error	0.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023010 
(FPCore (x)
  :name "bug333 (missed optimization)"
  :precision binary64
  :pre (and (<= -1.0 x) (<= x 1.0))

  :herbie-target
  (/ (* 2.0 x) (+ (sqrt (+ 1.0 x)) (sqrt (- 1.0 x))))

  (- (sqrt (+ 1.0 x)) (sqrt (- 1.0 x))))